1549301742-The_Theory_of_Difference_Schemes__Samarskii

476 Hmnogeneous Difference Schemes for Time-Dependent Equations

where fi±o = f(xi±o' t), or

Xi+O 5

( 48)

0 1

'Pi ='Pi = r;

z

j f(x,t)dx.

Xi-0.5

In 01·der to clarify the inain idea behind this approach, we turn to the

scheme with weights

(49) y 1 =A(0"1/+(l-O")y)+'P, xE...,.,· 1 ,, O<t=jr<T,

y(x, 0) = y 0 (x), x E wh, y(O, t) = u 1 (t), y(l, t) = u 2 (t),

w hosp augn1ented form is suitable for subsequent calculations. The elirni-

nation method unveils its potential once again and pennits us to find yj+l

on every new layer t = t j + 1 in terms of a known value yj on the current

layer t = tj:

A; ili-1 - C1f;; +Bi Yi+I = -F;, i = 1,2, ... ,N -1,

(}a. T

Ai= h ~ ,

1 z

Fi =

(1-O") T [ai Yi-1 ai+l Yi+r]

+ Ii z h 1 +! i,. + l + 'Pi.

Under such a choice of the computational algorithm the accuracy of
scheme ( 49) will be given special investigation. We are going to show that
it converges uniformly with the rate O( h^2 + r"'u) in the case of s1nooth
functions k(;r) and f(x,t).
For this, we proceed as usual. This amounts to evaluating the error
z{ = yf - u{, where Yi is a solution of the original pro blern ( 1)-(3) and